The long-range interactions among two- or three-atom systems are of considerable importance in the cold and ultracold research areas for many-body systems. For an ion and an atom, the long-range interaction potential is dominated by the induction (or polarization) potential resulting from the (classical) effect of the ion's electric field on the atom and the leading term of the induction potential is much stronger than the (quantum mechanical) dispersion (or van der Waals) interaction. The present paper focuses on the long-range interaction of the Li (2 2S) -Li (2 2S) -Li(1 1S) system, to see what changes this induction effect (originating in the electric field of the Li ion) yields in the long-range additive and nonadditive interactions of this three-body system. Using perturbation theory for energies, we evaluate the coefficients Cn in the potential energy for the three well-separated constituents, where n refers to the corresponding order in inverse powers of distance, obtaining the additive interaction coefficients C4, C6, C7, C8, C9 and the nonadditive interaction coefficients C7, C9. The obtained coefficients Cn are calculated with highly accurate variationally generated nonrelativistic wave functions in Hylleraas coordinates. Our calculations may be of interest for the study of three-body recombination and for constructing precise potential energy surfaces. We also provide precise evaluations of the long-range potentials for the two-body Li (2 2S) -Li(1 1S) system. For both the two-body and three-body cases, we provide results for the like-nuclei cases of 6Li and 7Li.